Gravity:

~Fgrav = −GM1M2

R212

r Ugrav = −GM1M2

R12

T =2πa3/2

√GM

Centre-of-mass:

~rcm =m1~r1 + m2~r2 + . . . + mn~rn

m1 + m2 + . . . + mn

(and similarly for ~v and ~a)

Forces: Newton’s:∑ ~F = m~a, ~FB on A = −~FA on B

Hooke’s: ~Felas = −k(r − requil)r

friction: |~fs| ≤ µs|~n|, |~fk| = µk|~n|

Constants/Conversions:

g = 9.80 m/s2

= 32.15 ft/s2

(on Earth’s surface)

G = 6.674 × 10−11 N · m2/kg2

R⊕ = 6.38 × 106 m M⊕ = 5.98 × 1024 kgR⊙ = 6.96 × 108 m M⊙ = 1.99 × 1030 kg

1 km = 0.6214 mi 1 mi = 1.609 km

1 ft = 0.3048 m 1 m = 3.281 ft

1 hr = 3600 s 1 s = 0.0002778 hr

1 kgms2

= 1 N = 0.2248 lb 1 lb = 4.448 N

1 J = 1 N·m 1 W = 1 J/s

1 rev = 360◦ = 2π radians 1 hp = 745.7 W

Relative velocity: ~vA/C = ~vA/B + ~vB/C

~vA/B = −~vB/A

Circular motion: |~arad| =v2

RT =

2πR

v

s = Rθ vtan = Rω atan = Rα

Forces, Energy and Momenta:

translational rotational

Ktrans = 12Mv2

cm

W =∫

~F · d~r const−−−→force

~F · ∆~r

P = dWdt = ~F · ~v

~pcm = m1~v1 + m2~v2 + . . .

= M~vcm

~J =∫

~Fdt = ∆~p

∑ ~Fext = M~acm =d~pcm

dt∑ ~Fint = 0

~τ = ~r × ~F and |~τ | = F⊥r

Krot = 12Itotω

2

W =∫

τdθconst−−−−→torque

τ ∆θ

P = dWdt = ~τ · ~ω

~L = I1~ω1 + I2~ω2 + . . .

= Itot~ω

= ~r × ~p

∑

~τext = Itot~α =d~L

dt∑

~τint = 0

—– Both translational and rotational —–

W = ∆K = Ktrans,f + Krot,f − Ktrans,i − Krot,i

Etot,f = Etot,i + Wother ⇔ Kf + Uf = Ki + Ui + Wother

U = −∫

~F · d~r ; Ugrav = Mgycm ; Uelas = 12k(r − requil)

2

Fx(x) = −dU(x)/dx ~F = −~∇U = −[

∂U∂x i + ∂U

∂y j + ∂U∂z k

]

Equations of motion:

translational rotational

—– constant (linear/angular) acceleration only —–

~r(t) = ~r◦ + ~v◦t + 12~at2

~v(t) = ~v◦ + ~at

v2x = v2

x,0 + 2ax(x − x0)(and similarly for y and z)

~r(t) = ~r◦ + 12(~vi + ~vf )t

θ(t) = θ◦ + ω◦t + 12αt2

ω(t) = ω◦ + αt

ω2f = ω2

◦ + 2α(θ − θ◦)

θ(t) = θ◦ + 12(ωi + ωf )t

———– always true ———–

〈~v〉 = ~r2−~r1

t2−t1~v = d~r

dt

〈~a〉 = ~v2−~v1

t2−t1~a= d~v

dt = d2~rdt2

~r(t) = ~r◦ +∫ t

0~v(t′) dt′

~v(t) = ~v◦ +∫ t

0~a(t′) dt′

〈ω〉 = θ2−θ1

t2−t1ω = dθ

dt

〈α〉 = ω2−ω1

t2−t1α= dω

dt = d2θdt2

θ(t) = θ◦ +∫ t

0~ω(t) dt

ω(t) = ω◦ +∫ t

0~α(t) dt

General math:

ha

hho

φ

θ

ha = h cos θ = h sin φ

ho = h sin θ = h cos φ

h2 = h2a + h2

o tan θ =ho

ha

~A = Axi + Ay j + Az k

~A · ~B = AxBx + AyBy + AzBz = AB cos θ = A‖B = AB‖

~A × ~B = (AyBz−AzBy )i + (AzBx−AxBz)j + (AxBy−AyBx)k

= AB sin θ = A⊥B = AB⊥

df/dt = natn−1

If f(t) = atn, then

∫ t2t1

f(t)dt = an+1

(tn+12 − tn+1

1 ) (for n 6= −1)∫

f(t)dt = an+1

tn+1 + C (for n 6= −1)

ax2 + bx + c = 0 ⇒ x =−b ±

√b2 − 4ac

2a

Phys 218 — Exam III Formulae

à For a point-like particle of mass M a distance R from the axis of rotation: I = MR2

à Parallel axis theorem: Ip = Icm + Md2